4,041 research outputs found
Recent Results on Balanced Symmetric Boolean Functions
In this paper we prove all balanced symmetric Boolean functions of fixed degree are trivial when the number of variables grows large enough. We also present the nonexistence of trivial balanced elementary symmetric Boolean functions except for and , where and are any positive integers, which shows Cusick\u27s conjecture for balanced elementary symmetric Boolean functions is exactly the conjecture that all balanced elementary symmetric Boolean functions are trivial balanced. In additional, we obtain an integer , which depends only on , that Cusick\u27s conjecture holds for any
Linear recurrences and asymptotic behavior of exponential sums of symmetric boolean functions
In this paper we give an improvement of the degree of the homogeneous linear
recurrence with integer coefficients that exponential sums of symmetric Boolean
functions satisfy. This improvement is tight. We also compute the asymptotic
behavior of symmetric Boolean functions and provide a formula that allows us to
determine if a symmetric boolean function is asymptotically not balanced. In
particular, when the degree of the symmetric function is a power of two, then
the exponential sum is much smaller than .Comment: 18 pages, 3 figure
Balanced Symmetric Functions over
Under mild conditions on , we give a lower bound on the number of
-variable balanced symmetric polynomials over finite fields , where
is a prime number. The existence of nonlinear balanced symmetric
polynomials is an immediate corollary of this bound. Furthermore, we conjecture
that are the only nonlinear balanced elementary symmetric
polynomials over GF(2), where , and we prove various results in support of this conjecture.Comment: 21 page
Fast Algebraic Attacks and Decomposition of Symmetric Boolean Functions
Algebraic and fast algebraic attacks are power tools to analyze stream
ciphers. A class of symmetric Boolean functions with maximum algebraic immunity
were found vulnerable to fast algebraic attacks at EUROCRYPT'06. Recently, the
notion of AAR (algebraic attack resistant) functions was introduced as a
unified measure of protection against both classical algebraic and fast
algebraic attacks. In this correspondence, we first give a decomposition of
symmetric Boolean functions, then we show that almost all symmetric Boolean
functions, including these functions with good algebraic immunity, behave badly
against fast algebraic attacks, and we also prove that no symmetric Boolean
functions are AAR functions. Besides, we improve the relations between
algebraic degree and algebraic immunity of symmetric Boolean functions.Comment: 13 pages, submitted to IEEE Transactions on Information Theor
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